I don't think it being one of Goku's techniques would make too much of a difference to Vegeta at this point, as he's already used the same training methods as Goku (eg. Gravity Chamber) and forsaken his pride to reach Goku's level (Majin Vegeta). The main reason I believe he doesn't do so is due to the stress Kaioken has on the body making it a highly inefficient technique for use against someone like Beerus or Whis.